A formal derivation of the convective amplification of a three-wave parametric instability (PI) in an inhomogeneous plasma is revisited. The confusion of the mixed use of the initial values of PI is clarified and the consistency among the initial values is discussed. Through the exact solutions of the coupling equations of a three-wave PI, it is suggested that the initial amplitude of low frequency plasma fluctuations might be the appropriate initial value for all the PI scenarios, especially in magnetic confinement fusion plasma. The co-effect of plasma inhomogeneity and finite pump profile is accordingly evaluated through the exact solutions of the coupling equations, where the effects of a finite pump profile are found to be totally different for forward scatters and backward scatters.
This paper revisits the formal derivation of the convective amplification of a three-wave parametric instability (PI) in an inhomogeneous plasma. It clarifies the confusion in the use of initial values for PI and suggests that the initial amplitude of low-frequency plasma fluctuations might be the appropriate initial value. The co-effect of plasma inhomogeneity and finite pump profile on PI is also evaluated.